On The Significance of Bell’s Locality Condition
A. Shafieea and M. Golshanib
Institute for studies in Theoretical Physics and Mathematics P.O. Box 19395-5531, Tehran, Iran
E-mail: ashafiee@theory.ipm.ac.ir, bgolshani@ihcs.ac.ir
ABSTRACT. Reviewing the general representation of a stochastic local hidden variables theory in the context of an ideal Bohm’s version of the EPR experiment, we show explicitly that the violation of Bell’s locality condition is due to the assumption of “outcome independence” at the hidden variables level. Also, we show that if we introduce determinism, the assumption of outcome independence will be allowed.
1 Introduction
Since the derivation of original Bell’s inequality [1], this inequality (es- pecially in the form of CH [2] & CHSH [3] inequalities) has been derived in various ways. It has been extended to more than two particles and to the cases where the initial state’s spin is higher than zero. But what is generally accepted as an interpretation of all forms of Bell’s inequality is that there exists an incompatibility between any local hidden vari- ables theory with quantum mechanical predictions, and this is due to non-locality.
Because of the central role of the locality condition in Bell’s theorem, there has been some efforts to clarify its meaning by considering other conceptual possibilities. One of the most influential works done in this context is due to Jarrett who obtained Bell’s locality condition by the conjunction of two independent conditions [4]. Shimony called the first condition “outcome independence“ [5], which states that in the Bohmian version [6] of the EPR experiment [7] (hereafter called EPRB), and for definite settings on both sides, the probability of getting a result on one side is independent of the result on the other side. The second condition called “parameter independence“ by Shimony, holds that the result of
a given measurement is statistically independent of the setting of the distant measuring apparatus.
We shall discuss these points in more detail in this paper, but what is important here is that the violation of Bell’s inequality implies that either parameter or outcome independence or both must fail , but it can not tell us which of the two conditions is violated. So far, no generally accepted solution has been found for this problem [8].
In section 2, we show explicitly that showing the representation of the stochastic local hidden variable theories as done by CH [2] and Bell et al. [9], there is no stochastic outcome independent hidden variables the- ory which can reproduce all the predictions of quantum mechanics. To show the inconsistency, we restrict ourselves to an ideal EPRB thought experiment and define a condition applicable at the quantum level to prove a fundamental relation (relation (19)) in the context of a hidden variables theory, without using any inequality. Then, in section 3, we show that the outcome independence condition can only be considered in the deterministic hidden variables theories.
2 Stochastic local hidden variables theories
Let us consider a system consisting of a source, which emits two spin 1/2 particles. The spin of particles 1 and 2 are measured in the directions b
a and bb, respectively. Ignoring the possibilities concerning the missed detections (as is the case for an ideal thought experiment), one can use quantum mechanics to calculate various probabilities. For example, the joint probability for the spin component of particle 1 in the directionba beingr(r=±1 in units of }
2 ) and the spin component of particle 2 in the directionbbbeing q(q=±1 in units of }
2) is equal to P(12)(σa =r, σb =q|ba,bb,Ψ0) =1
4(1−rqcosθab) (1) where Ψ0is the singlet state representing the initial state wave function andθab is the angle betweenbaandbb .
Noticing the relation between the average values and the probability measures, one can write the joint probability defined in (1) as [10], P(12)(σa=r, σb=q|ba,bb,Ψ0) =1
4[1 +r < σa >+q < σb>+rq < σaσb>]
(2)
Here,< σa >is the expectation value of the spin component of particle 1 alongba,< σaσb>is the expectation value of the product of the values of the spin components of particles 1 and 2 alongbaandbb, respectively, and< σb>is the expectation value of the spin component of particle 2 alongbb.
Summing overqorrin (2), yields marginal probability measures for particle 1 or 2, respectively,
P(1)(σa=r|ba,Ψ0) = 1
2[1 +r < σa>] (3) and
P(2)(σb=q|bb,Ψ0) =1
2[1 +q < σb>] (4) To have a realistic interpretation of quantum mechanics we seek a stochastic local hidden variables theory [2, 9] in which λ represents a collection of hidden variables which determine the complete state of the particles and which belongs to a space Λ on which the probability mea- sures can be defined. Here, one can assume that a spin component of a particle has a definite value even before it is measured, and that the statistical results of the spin measurements of one particle do not depend on the values of the spin components of the other particle. Thus, if the spin component of particle 1 is measured first, we have
p(2)(σb=q|ba,bb, σa=r, λ) =p(2)(σb=q|ba,bb, λ) (5) where p is the symbol of the probability measures on space Λ. This relation is equivalent to outcome independence. According to (5), the probability for particle 2 having spin component along bb equal to q is independent of the value of the spin component of particle 1 alongba, if the complete stateλand the distant settings are given. Furthermore, if one assumes Shimony’s parameter independence, one gets
p(12)(σa=r, σb=q|ba,bb, λ) =p(1)(σa=r|ba, λ)p(2)(σb =q|bb, λ)
= 1 4 h
1 +rE(1)(ba, λ) +qE(2)(bb, λ) +rqE(12)(ba,bb, λ) i
(6) where,
p(1)(σa =r|ba, λ) = 1 2 h
1 +rE(1)(ba, λ) i
(7) and
p(2)(σb =q|bb, λ) =1 2 h
1 +qE(2)(bb, λ) i
(8) Here, E(1)(ba, λ) and E(2)(bb, λ) are, respectively, the average values of the spin components of particle 1 along ba and particle 2 along bb , and E(12)(ba,bb, λ) =E(1)(ba, λ)E(2)(bb, λ) represents the average value of the product of the spin components of particles 1 and 2 alongba andbb, respectively. The relation (6) is known as Bell’s locality condition.
Multiplying E(12)(ba,bb, λ) through the probability density ρ(λ) and integrating overλ(R
Λρ(λ)dλ= 1 ), we get the corresponding expecta- tion value at the quantum level, i.e.,
< σaσb>=
Z
Λ
E(12)(ba,bb, λ)ρ(λ)dλ (9) Similar relations hold for hσai and hσbi, regarding E(1)(ba, λ) and E(2)(bb, λ), respectively.
As a consequence of the consistency conditions in the laws of proba- bility, we also have for the conditional probability of the second particle [11],
P(2)(σb=q|ba,bb, σa=r,Ψ0) = R
Λp(1)(σa =r|ba, λ)p(2)(σb=q|bb, λ)ρ(λ)dλ P(1)(σa =r|ba,Ψ0)
(10) What we want to show explicitly now is that the foregoing framework for the representation of a stochastic local hidden variables theory is not successful, even if we accept that superluminal influences do not exist.
To clarify this matter, we look at some of the fundamental require- ments which are relevant to Bell-type ideal experiments. First, we con- sider two dynamical variables S1 and S2 which take values s1, s2 =± 1. Then, we define the exchangeability condition on s1 and s2 for the conditional probability measures as [12],
P(S2=s2|S1=s1) =P(S2=s1|S1=s2) (11) Now, one can show that by imposing this condition, we have the following result (see appendix):
P(S2=s2|S1=s1) =1
2[1 +s1s2< S1S2>] (12) Under the conditions assumed in the EPRB problem (i.e., hσai = hσbi= 0) if we defineS1 ≡σa andS2 ≡σb (when one definesS1≡σb
andS2≡σa, the argument remains valid for the probability measure of the first particle), the relation (12) takes the following form
P(2)(σb=q|ba,bb, σa =r,Ψ0) =1
2[1 +rq < σaσb>] (13) where< σaσb >=−cosθab.
According to (9), < σaσb >has a clear physical interpretation in a stochastic (outcome independent or dependent) hidden variables theory and r as well as q are completely defined in this framework. Thus, the right hand side of (13) is completely defined in a stochastic hidden variables theory and can be represented as a new relation,
f(r, q|ba,bb, λ) = 1 2 h
1 +rq E(12)(ba,bb, λ) i
(14) wheref(r, q|ba,bb, λ) satisfies the following relation,
P(2)(σb =q|ba,bb, σa =r,Ψ0) = Z
Λ
f(r, q|ba,bb, λ)ρ(λ)dλ (15)
Using the definition of E(12)(ba,bb, λ) as
E(12)(ba,bb, λ) = X
r,q=±1
rq p(12)(σa=r, σb=q|ba,bb, λ) and substituting it into (14), one obtains
f(r, q|ba,bb, λ) =p(12)(σa=r, σb=q|ba,bb, λ) +p(12)(σa=−r, σb=−q|ba,bb, λ) (16) This shows that f(r, q|ba,bb, λ) is a probability measure on the space Λ.
One can also deduce that
f(r, q|ba,bb, λ) +f(r,−q|ba,bb, λ) = 1 (17) Relation (17) shows thatf(r, q|ba,bb, λ) is a two-variable probability mea- sure, so that if we sum over one variable and fix the other, the result should lead to unity.
There is a subtle point that necessitates using an additional as- sumption in the relation (16). Let us consider, e.g., the functions f(+1,+1|ba,bb, λ) and f(+1,−1|ba,bb, λ). According to the relation (17), these functions are the probability measures whose values are defined in a set of the ideal experiments in which the overall number of de- tections should contain all the results (+1,+1), (+1,−1), (−1,+1) and (−1,−1) for r and q, respectively. If the overall number of detections reduces to the first two results (+1,+1) and (+1,−1), the functions f(+1,+1|ba,bb, λ) and f(+1,−1|ba,bb, λ) cannot be specified in principle.
This, in itself, makes no problem. The problem arises when one takes the relation (15) into account. When only the two-fold detections (+1,+1) and (+1,−1) are considered, the left hand side of the relation (15) is completely defined forq= +1 (or−1) andr= +1. This shows that one should be able to determine the function f(r, q|ba,bb, λ) for the two-fold detections too, when the variablesrand qare specified. The only kind of a two-variable probability measure f(r, q|ba,bb, λ) which satisfies both the relations (15) (which necessitates thatf should be defined in a two- fold probability space) and (17) (which is the normalization condition forf(r, q|ba,bb, λ)) is a conditional one. Thus, we conclude that
f(r, q|ba,bb, λ) =p(2)(σb=q|ba,bb, σa =r, λ) (18) Replacing (18) in (14), we get
p(2)(σb=q|ba,bb, σa =r, λ) = 1 2 h
1 +rq E(12)(ba,bb, λ) i
(19) This shows that the relation (5) cannot hold if the predictions of a stochastic hidden variables theory are to coincide with the results of the standard quantum mechanics, as is the case in relation (15). Conse- quently, we are confronted with the breakdown of outcome independence condition.
The relation (19) implies thatE(2)(ba,bb, λ) =E(1)(ba, λ)E(12)(ba,bb, λ) (see appendix). One can also show thatE(1)(ba,bb, λ) =E(2)(ba, λ)E(12)(ba,bb, λ), if we choose S1 ≡σb and S2 ≡σa, in relation (12). In an ideal EPRB experiment, however, the mean valuesE(1), E(2) andE(12) should not depend on our decision about which particle’s spin is measured first.
Thus, one can conclude thatE(1)=E(2)= 0 for everyba,bbandλ(E(12) cannot be taken to be always zero, because of the relation (9)).
The relation (19) depends on ba, and this is an indication of a kind of influence of the experimental arrangement for the measurement of the spin component of particle 1 along ba, on the particle 2. Such an influence is only meaningful on the basis of the violation of outcome in- dependence. Shimony has argued that the foregoing violation would not necessarily allow superluminal signals to be sent, and there remains a possibility of subluminal signaling in an outcome dependent stochastic hidden variables theory [5]. The inference that Bell’s theorem does not lead to the superluminal signalling has been analyzed by some people be- fore [13-16]. Here, we have reached the same conclusion by a completely different approach.
3 Deterministic hidden variables theories
There exists an exceptional situation, for which the exchangeability con- dition cannot be inferred from (19). This possibility occurs whenr and qare constrained by some deterministic conditions.
Assuming determinism in an outcome independent hidden variables theory (as is the case in Bohm’s theory), the knowledge of λ deter- mines the values of each particle’s spin along any direction. Thus, the
probability measures are either zero or one, and (regardless of consider- ing parameter independence) one can use definite values A(ba,bb, λ) and B(ba,bb, λ) instead of the average valuesE(1)(ba,bb, λ) andE(2)(ba,bb, λ), re- spectively. Then, if the spin components of both particles take the values
±1 in a deterministic way, we have
r A(ba,bb, λ) = +1 and q B(ba,bb, λ) = +1 f or everyba,bb and λ. (20) In this situation, one can deduce that p(1)DET(σa =r|ba,bb, λ) is equal to one forr A(ba,bb, λ) = +1, and zero forr A(ba,bb, λ) =−1, which means the last case never occurs. A similar result holds for p(2)DET(σb =q|ba,bb, λ).
Here, r and q are constrained by A(ba,bb, λ) andB(ba,bb, λ), respectively.
Consequently, the relation (19) should be replaced by the following re- lation,
p(2)DET(σb=q|ba,bb, σa=r, λ) =1 2 h
1 +rq A(ba,bb, λ)B(ba,bb, λ) i
=fDET(r, q|ba,bb, λ) (21) which is obtained by assuming thatEDET(12) (ba,bb, λ) =A(ba,bb, λ)B(ba,bb, λ), without introducing the parameter independence condition. Since, the situation q A(ba,bb, λ) = +1 is only allowed when q =r, the exchange- ability condition cannot be established here for r6= q. Also, since the situation r A(ba,bb, λ) = −1 never occures, the relation (21) reduces to the following form,
p(2)DET(σb=q|ba,bb, σa=r, λ) =1
2[1 +q B(ba, λ)]
=p(2)DET(σb=q|ba,bb, λ) (22) which is equal to (5), if we introduce the assumption of determinism.
Here, the function fDET(r, q|ba,bb, λ) cannot reproduce the conditional
probability P(2)(σb = q|ba,bb, σa = r,Ψ0) in relation (15), and the ex- changeability condition at the quantum level does not contradict the out- come independence condition at the hidden variables level. This shows thatthe outcome independence condition is only feasible in a determinis- tic framework,a situation in which the exchangeability condition is lost at the hidden variables level.
4 Appendix
To prove (12), we introduce two dynamical variables S1 and S2 taking valuess1,s2 =±1. One can express the joint probabilities in terms of some mean values (denoted byS1,S2 andS1S2),
P(S1=s1, S2=s2) = 1 4
£1 +s1S1+s2 S2+s1s2 S1S2
¤ (A.1)
The average quantities Si (i = 1,2) and S1S2 were represented at the quantum level by hSiiandhS1S2iand at the hidden variables level byE(i)and E(12), respectively. The marginal probabilityP(S1=s1) is obtained by summing overs2 in (A.1),
P(S1=s1) = 1 2
£1 +s1 S1¤
(A.2) Now, we can write (A.1) in the following form
P(S1=s1, S2=s2) = 1 4
£1 +s1S1¤ ·
1 +s2 S2+s1s2S1S2 1 +s1 S1
¸
(A.3) Using Bayesian probability and comparing (A.3) with (A.2), we get the conditional probabilityP(S2=s2|S1=s1) as
P(S2=s2|S1=s1) = 1 2
·
1 + s2 S2+s1s2 S1S2 1 +s1 S1
¸
(A.4) Exchangings1 ands2 leads to the following relation
P(S2=s1|S1=s2) = 1 2
·
1 + s1 S2+s1s2 S1S2 1 +s2 S1
¸
(A.5) Now if we impose the exchangeability condition (11) on (A.4) and (A.5), we get (12) (considering that s21 = s22 = 1), provided S2 = S1 S1S2.
In an ideal EPRB thought experiment, the exchangeability condition (11) holds because S2 = S1 = 0 (S1 =σa, S2 = σb). There are some examples for which the relationS2=S1S1S2 holds nontrivially(see e.g.
ref. [17]).
References
[1] J. S. Bell,Physics,1, 195(1964).
[2] J. F. Clauser, M. A. Horne,Phys. Rev. D 10, 526(1974).
[3] J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt,Phys. Rev. Lett.23, 80(1969).
[4] J. Jarrett,Nous,18, 569(1984).
[5] A. Shimony, in Sixty-Two Years of Uncertainty: Historical, Philosophi- cal, and Physical Inquiries into the Foundations of Quantum Mechanics, edited by A. Miller (Plenum, New York, 1990), pp. 33-43.
[6] D. Bohm,Quantum Theory (Prentice-Hall, Englewood cliffs, NJ, 1951), pp.614-619.
[7] A. Einstein, B. Podolsky, and N. Rosen,Phys. Rev.41,777(1935).
[8] T. Madlin,Quantum Non-locality and Relativity (Blackwell Publishers, Society Series, Vol. 13, Cambridge, 1994), pp. 90-96.
[9] J. S. Bell, A. Shimony, M. A. Horne and J. F. Clauser, Dialectica, 39, 85(1985).
[10] A. Fine,Phys. Rev. Lett.48, 291(1982).
[11] A. J. M. Garrett,Found. Phys.20, 381(1990).
[12] P. Suppes and M. Zanotti, in Studies in the Foundations of Quantum Mechanics, edited by P. Suppes (Philosophy of Science Association, East Lansing, Michigan, 1980), pp. 173-191.
[13] W. De Baere, A. Mann and M. Revzen,Found. Phys.29, 67(1999).
[14] L. E. Szabo,Found. Phys. Lett.8, 417(1995).
[15] L. de Broglie, G. Lochak, J. A. Beswick and J. Vassalo-Pereira,Found.
Phys.6, 3(1976).
[16] G. Lochak, Found. Phys. 6,173(1976); G. Lochak, inThe Wave-Particle Dualism, edited by S. Diner, D. Fargue, G. Lochak and F. Selleri (D.
Reidal Publishing Co., Dordrecht, 1984), pp. 1-25.
[17] A. Shafiee and M. Golshani,Single-particle Bell-type Inequality, submit- ted toAnn. Fond. L. de Broglie.
(Manuscrit re¸cu le 16 avril 2001)
